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Simplifying 0 = 6.28r2 + 215073.2r + -25722.88 Reorder the terms: 0 = -25722.88 + 215073.2r + 6.28r2 Solving 0 = -25722.88 + 215073.2r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 25722.88 = 25722.88 25722.88 + -215073.2r + -6.28r2 = -25722.88 + 215073.2r + 6.28r2 + 25722.88 + -215073.2r + -6.28r2 Reorder the terms: 25722.88 + -215073.2r + -6.28r2 = -25722.88 + 25722.88 + 215073.2r + -215073.2r + 6.28r2 + -6.28r2 Combine like terms: -25722.88 + 25722.88 = 0.00 25722.88 + -215073.2r + -6.28r2 = 0.00 + 215073.2r + -215073.2r + 6.28r2 + -6.28r2 25722.88 + -215073.2r + -6.28r2 = 215073.2r + -215073.2r + 6.28r2 + -6.28r2 Combine like terms: 215073.2r + -215073.2r = 0.0 25722.88 + -215073.2r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2 25722.88 + -215073.2r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 25722.88 + -215073.2r + -6.28r2 = 0.00 Begin completing the square. Divide all terms by -6.28 the coefficient of the squared term: Divide each side by '-6.28'. -4096 + 34247.32484r + r2 = 0 Move the constant term to the right: Add '4096' to each side of the equation. -4096 + 34247.32484r + 4096 + r2 = 0 + 4096 Reorder the terms: -4096 + 4096 + 34247.32484r + r2 = 0 + 4096 Combine like terms: -4096 + 4096 = 0 0 + 34247.32484r + r2 = 0 + 4096 34247.32484r + r2 = 0 + 4096 Combine like terms: 0 + 4096 = 4096 34247.32484r + r2 = 4096 The r term is 34247.32484r. Take half its coefficient (17123.66242). Square it (293219814.7) and add it to both sides. Add '293219814.7' to each side of the equation. 34247.32484r + 293219814.7 + r2 = 4096 + 293219814.7 Reorder the terms: 293219814.7 + 34247.32484r + r2 = 4096 + 293219814.7 Combine like terms: 4096 + 293219814.7 = 293223910.7 293219814.7 + 34247.32484r + r2 = 293223910.7 Factor a perfect square on the left side: (r + 17123.66242)(r + 17123.66242) = 293223910.7 Calculate the square root of the right side: 17123.78202092 Break this problem into two subproblems by setting (r + 17123.66242) equal to 17123.78202092 and -17123.78202092.Subproblem 1
r + 17123.66242 = 17123.78202092 Simplifying r + 17123.66242 = 17123.78202092 Reorder the terms: 17123.66242 + r = 17123.78202092 Solving 17123.66242 + r = 17123.78202092 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-17123.66242' to each side of the equation. 17123.66242 + -17123.66242 + r = 17123.78202092 + -17123.66242 Combine like terms: 17123.66242 + -17123.66242 = 0.00000 0.00000 + r = 17123.78202092 + -17123.66242 r = 17123.78202092 + -17123.66242 Combine like terms: 17123.78202092 + -17123.66242 = 0.11960092 r = 0.11960092 Simplifying r = 0.11960092Subproblem 2
r + 17123.66242 = -17123.78202092 Simplifying r + 17123.66242 = -17123.78202092 Reorder the terms: 17123.66242 + r = -17123.78202092 Solving 17123.66242 + r = -17123.78202092 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-17123.66242' to each side of the equation. 17123.66242 + -17123.66242 + r = -17123.78202092 + -17123.66242 Combine like terms: 17123.66242 + -17123.66242 = 0.00000 0.00000 + r = -17123.78202092 + -17123.66242 r = -17123.78202092 + -17123.66242 Combine like terms: -17123.78202092 + -17123.66242 = -34247.44444092 r = -34247.44444092 Simplifying r = -34247.44444092Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.11960092, -34247.44444092}
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